Find the vector equation of the line passing through point A(1, 2, 3) with direction vector d = (2, -1, 4).
Answer and explanation
Correct answer: A. r = (1, 2, 3) + t(2, -1, 4)
- Step 1: The vector equation of a line in 3D space is r = r_0 + t * d, where r_0 is the position vector of a point on the line and d is the direction vector.
- Step 2: Substitute r_0 = (1, 2, 3) and d = (2, -1, 4): r = (1, 2, 3) + t(2, -1, 4).
Answer
r = (1, 2, 3) + t(2, -1, 4).
Why not the others: 'r = (2, -1, 4) + t(1, 2, 3)' swaps point and direction vector; 'r = (1, 2, 3) . (2, -1, 4)' is a scalar dot product; '2x - y + 4z = 12' represents a plane, not a line.
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